Yesterday, 05:44 PM
(This post was last modified: Yesterday, 05:47 PM by npcompl33t.)
ok, several questions here, let me know if i missed anything.
Do you use raw frequencies or Bayesian estimates?
Raw frequencies, without pseudo counts or Bayesian smoothing. I estimate a separate distribution of following glyphs for each starting glyph.
P(h\ |g)= C(g,h) \ sum_j C(g,j)
I then combine the JS divergences between these conditional distributions. A row contributes only if it has at least 3 transitions in each profile. Excluded weights are not redistributed to the remaining rows.
How does JS distance compare with Euclidean distance near the center
JS divergence is different than JS distance. JS distance is the square root of JS divergence. Near the center, small JS distances are approximately a constant multiple of Euclidean distances.
What changes near a vertex?
The factor 1/p makes changes involving rare categories more influential. There is not a simple conversion factor, it depends on the movement direction. Introducing a previously absent category can produce a larger JS change relative to Euclidean. JS is finite but it is sensitive near zero probabilities.
What are its geodesics?
There is a distinction between direct distance between two points or endpoints, and the length accumulated while travelling through intermediate distributions. Ordinary JS distance is generally not the shortest path along a simplex. Its infintesimal geometry is proportional to the Fisher-Rao metric. If we use local geometry to measure continuous path lengths, shortest paths become great circle arcs after transforming coordinates to sqrt(p_i). When mapped back to probabilities, they are generally curved. The curves i fitted above using ElPiGraph are fitted principle curves in the MDS representation, NOT JS geodesics.
Can JS distance be represented as Euclidean DIstance?
Yes, in Hlibert space, but not generally in R^D or R^D-1. In my model, pair dependent weights and row exclusions mean it is not ordinary JS on one fixed probability vector. It is probably better described as weighted JS dissimilarity, I do not claim all the guarantees of ordinary JS distance.
If i am trying to get to the heart of what you are asking with this line of inquiry, my guess would be something like: does the horseshoe and disputed ordering persist when sampling zeros/sparse rows/probability estimation is handled differently?
Is that accurate? Because I believe i can answer that question, but I would want to agree with you ahead of time on the actual test and what the results would indicate -- you still have not gotten back to me on the last test you suggested that I ran.
Do you use raw frequencies or Bayesian estimates?
Raw frequencies, without pseudo counts or Bayesian smoothing. I estimate a separate distribution of following glyphs for each starting glyph.
P(h\ |g)= C(g,h) \ sum_j C(g,j)
I then combine the JS divergences between these conditional distributions. A row contributes only if it has at least 3 transitions in each profile. Excluded weights are not redistributed to the remaining rows.
How does JS distance compare with Euclidean distance near the center
JS divergence is different than JS distance. JS distance is the square root of JS divergence. Near the center, small JS distances are approximately a constant multiple of Euclidean distances.
What changes near a vertex?
The factor 1/p makes changes involving rare categories more influential. There is not a simple conversion factor, it depends on the movement direction. Introducing a previously absent category can produce a larger JS change relative to Euclidean. JS is finite but it is sensitive near zero probabilities.
What are its geodesics?
There is a distinction between direct distance between two points or endpoints, and the length accumulated while travelling through intermediate distributions. Ordinary JS distance is generally not the shortest path along a simplex. Its infintesimal geometry is proportional to the Fisher-Rao metric. If we use local geometry to measure continuous path lengths, shortest paths become great circle arcs after transforming coordinates to sqrt(p_i). When mapped back to probabilities, they are generally curved. The curves i fitted above using ElPiGraph are fitted principle curves in the MDS representation, NOT JS geodesics.
Can JS distance be represented as Euclidean DIstance?
Yes, in Hlibert space, but not generally in R^D or R^D-1. In my model, pair dependent weights and row exclusions mean it is not ordinary JS on one fixed probability vector. It is probably better described as weighted JS dissimilarity, I do not claim all the guarantees of ordinary JS distance.
If i am trying to get to the heart of what you are asking with this line of inquiry, my guess would be something like: does the horseshoe and disputed ordering persist when sampling zeros/sparse rows/probability estimation is handled differently?
Is that accurate? Because I believe i can answer that question, but I would want to agree with you ahead of time on the actual test and what the results would indicate -- you still have not gotten back to me on the last test you suggested that I ran.